Having established regularity of the universal moduli space, we can apply Sard–Smale to the projection \(\widetilde{\mathcal {M}}^{q,p,*}_{\lbrace \Gamma _i\rbrace }(\mathbf {y},A,\mathsf {t},\mathbb {P}^s)|_C \rightarrow \mathbb {P}^s\) .
We conclude that \(\widetilde{\mathcal {M}}^{q,p,*}_{\lbrace \Gamma _i\rbrace }(\mathbf {y},A,\mathsf {t},\mathbf {P})|_C\)  (and hence, by definition, \(\mathcal {M}^{q,p,*}_{\lbrace \Gamma _i\rbrace }(\mathbf {y},A,\mathsf {t},\mathbf {P})|_C\) ) is regular for a comeagre subset of the set of choices of \(C^s\)  perturbation data \(\mathbb {P}^s\) , for each \(C\) .
It follows by an argument due to Taubes (as in [1]}) that \(\mathcal {M}^*_{\lbrace \Gamma _i\rbrace }(\mathbf {y},A,\mathsf {t},\mathbf {P})|_C\)  is regular for a comeagre subset of the set of choices of smooth perturbation data. Taking the intersection over all \(C\) , we conclude that \(\mathcal {M}^*_{\lbrace \Gamma _i\rbrace }(\mathbf {y},A,\mathsf {t},\mathbf {P})\)  is regular for a comeagre subset of the set of smooth perturbation data, as required.
